GIFT  OF 


The  Distribution  of  Current  and  the  Variation  of 

Resistance  in  Linear  Conductors  of  Square  and 

Rectangular  Cross-Section  when  Carrying 

Alternating  Currents  of  High  Frequency 


BY 

HIRAM   WHEELER  EDWARDS 


A  DISSERTATION 

IN  PARTIAL  SATISFACTION  OF  THE 

REQUIREMENTS  FOR  THE  DEGREE  OF  DOCTOR  OF  PHILOSOPHY 

SUBMITTED  TO  THE  FACULTY  OF  THE 

COLLEGE  OF  NATURAL  SCIENCES 

UNIVERSITY  OF  CALIFORNIA 


MAY  i,  1911 


The  Distribution  of  Current  and  the  Variation  of 

Resistance  in  Linear  Conductors  of  Square  and 

Rectangular  Cross-Section  when  Carrying 

Alternating  Currents  of  High  Frequency 


BY 


HIRAM   WHEELER  EDWARDS 

\\ 


A  DISSERTATION 

IN  PARTIAL  SATISFACTION  OF  THE 

REQUIREMENTS  FOR  THE  DEGREE  OF  DOCTOR  OF  PHILOSOPHY 

SUBMITTED  TO  THE  FACULTY  OF  THE 

COLLEGE  OF  NATURAL  SCIENCES 

UNIVERSITY  OF  CALIFORNIA 


MAY  i,  1911 


[Reprinted  from  the  PHYSICAL  REVIEW,  Vol.  XXXIII. ,  No.  3;  September ,;r£i£.j;  **£  j 


THE   DISTRIBUTION   OF   CURRENT   AND   THE  VARIATION 

OF  RESISTANCE  IN  LINEAR  CONDUCTORS  OF  SQUARE 

AND   RECTANGULAR  CROSS-SECTION  WHEN 

CARRYING  ALTERNATING  CURRENTS  OF 

HIGH  FREQUENCY. 

BY  HIRAM  WHEELER  EDWARDS. 

I.  INTRODUCTION. 

HHE  present  paper  undertakes  the  investigation  of  the  virtual  resist- 
•*•  ance,  inductance  and  current  distribution  of  long,  straight  conduc- 
tors of  square  and  rectangular  cross-section,  when  carrying  alternating  cur- 
rents of  high  frequency.  Experimental  observations  are  given,  which, 
within  certain  limits,  corroborate  the  calculations  for  virtual  resistance, 
but  no  attempt  was  made  to  measure  the  current  intensity  and  inductance. 
Approximate  formulas  are  derived  by  means  of  which  the  virtual  resist- 
ance, the  inductance  and  the  current  intensity  may  be  calculated  for 
particular  cases.  .  v, 

The  problem  of  determining  the  virtual  resistance  and  inductance  of 
long,  straight,  isolated  conductors,  when  carrying  alternating  current 
is  not  a  new  one.  In  the  oscillating  current  circuit  of  wireless  telegraph, 
the  increase  of  resistance  due  to  the  high  frequencies  of  alternation  is  a 
very  impotrant  factor.  The  Tesla  experiments  illustrating  the  "skin- 
effect"  show  that  in  a  number  of  cases  the  virtual  resistance  is  many 
times  the  resistance  offered  to  direct  current.  In  the  alternating-cur- 
rent railroads,  the  use  of  the  rails  for  the  return  circuit  is  impracticable 
because  of  the  increased  resistance.1 

For  long,  straight,  cylindrical  wires,  Maxwell2  has  developed  an  expres- 
sion for  the  virtual  resistance  and  inductance.  Rayleigh3  has  modified 
Maxwell's  results  by  showing  that  the  permeability  of  the  material  of 
the  conductor,  if  greater  than  unity,  causes  a  further  increase  in  the  resist- 
ance. Kelvin4  solved  the  problem  in  a  different  manner,  obtaining 
results  similar,  in  final  form,  to  those  of  Maxwell.  He  has  given  simpli- 
fied forms  of  expression  for  virtual  resistance  and  inductance  for  the 

^Standard  Handbook,  for  El.  Eng.,  Sec.  2,  No.  122;  Sec.  n,  No.  33. 

2  Maxwell,  E.  &  M.,  Vol.  2,  third  ed.,  p.  320.     See  also  Gray,  E.  &.  M.f  Vol.  2,  Pt.  I,  p.  325 ; 
and  Thomson,  Recent  Researches  in  E.  &.  M.,  Chap.  4. 

3  Rayleigh,  Phil.  Mag.,  Vol.  21,  1886,  p.  381. 

<  Kelvin,  Math.  &  Phys.  Papers,  Vol.  3,  p.  462. 


244550 


'1,5'..  HIRAM  WHEELER  EDWARDS.  [VOL.  XXXIII. 

particular  cases  of  low  and  high  frequency.  In  a  recent  paper,  Nicholson1 
has  published  another  solution  of  this  problem.  He  considers  first  two 
parallel  conductors  carrying  equal  currents  in  opposite  directions  and 
then  arrives  at  the  case  of  the  single  wire  by  neglecting  the  effect  of  the 
second  upon  it.  In  a  paper  on  the  "  Effective  Resistance  and  Inductance 
of  a  Concentric  Main,"  Russell2  has  given  formulas  for  the  virtual 
resistance  and  inductance  of  a  concentric  main  with  a  solid  inner  con- 
ductor. The  expression  for  the  virtual  resistance  is  conveniently  arranged 
so  that  the  resistance  of  either  the  core  or  the  sheath  may  be  calculated  for 
any  particular  case.  He  shows  how  the  formulas  become  simplified 
for  the  cases  of  direct  current  and  alternating  currents  of  low  and  high 
frequency.  A  consideration  of  the  current  density  in  the  inner  and  outer 
conductors  of  the  main  is  another  feature  of  this  paper. 

Mordey3  has  employed  Kelvin's  formulas  in  the  numerical  computation 
of  the  increase  of  resistance  of  a  cylindrical  conductor,  when  carrying 
alternating  currents  of  commercial  frequency,  80  to  133  complete  cycles 
per  second.  Thus  with  a  frequency  of  80,  the  resistance  of  a  copper 
conductor,  2.5  cm.  in  diameter,  increases  17.5  per  cent.  The  same  per- 
centage increase  is  computated  for  a  conductor  of  2.24  cm.  at  100  cycles 
per  second,  and  also  for  a  conductor  of  1.036  cm.  at  133  cycles  per  second. 

Fleming4  has  measured  the  ratio  of  the  virtual  to  the  direct  current 
resistance  of  cylindrical  wires  using  alternating  currents  of  frequency 
somewhat  less  than  half  a  million  per  second.  For  example,  with  a  fre- 
quency of  440,000  using  a  copper  wire,  no.  14  standard  wire  gauge,  he 
finds  the  ratio  to  be  5.46  by  Russell's  formula  while  his  measured  value 
is  5.90.  He  considers  this  agreement  between  the  values,  which  differ 
by  about  eight  per  cent.,  as  satisfactory.  Values  are  given  below  for 
cylindrical  wires  which  differ  by  less  than  one  per  cent.,  and  for  square 
and  rectangular  sectioned  wires  about  four  per  cent.  The  experimental 
work  on  cylindrical  wires,  cited  below,  was  performed  with  slightly 
damped  oscillating  currents,  to  determine  whether  the  damping  was  great 
enough  to  make  any  appreciable  difference  between  the  results  observed 
from  damped  currents  and  calculated  by  a  formula  derived  upon  the 
assumption  that  the  currents  were  undamped.  Fleming  used  undamped 
oscillations  in  obtaining  his  values. 

2.  EXPERIMENTAL  METHOD. 

(a)  Production  of  Currents  of  High  Frequency. — The  two  most  common 
methods  of  producing  alternating  currents  of  high  frequency  are  the 

1  Nicholson,  Phil.  Mag.,  Vol.  17,  1909,  p.  255. 

2  Russell,  Phil.  Mag.,  Vol.  17,  1909,  p.  524. 

3  Mordey,  Electrician,  May  31,  1889,  p.  94. 
4Fleming,  Electrician,  December  17,  1909,  p.  38  i. 


0.3-] 


THE  DISTRIBUTION  OF  CURRENT. 


186 


i.e. 


singing-arc  method  and  by  the  discharge  of  a  condenser  through  an 
inductive  resistance.  Both  methods  were  tried  but  greater  success  was 
obtained  by  the  latter,  so  it  was  adopted  in  the  experimental  work  herein 
recorded.  The  oscillations  of  current  from  a  condenser  are  more  or  less 
damped,  but  the  damping  was  reduced  to  such  a  magnitude,  in  the 
circuits  here  described,  that  the  results  did  not  appreciably  differ  from 
those  which  would  have  been  obtained  from  using  undamped  currents. 

The  scheme  of  connections  is  given  diagramatically  in  Fig.  I.  T.C. 
is  a  large  six-inch  induction 
coil .  I  ts  primary  is  connected 
to  a  source  of  direct-current 
supply.  I  is  a  Cunningham 
mercury-jet  interrupter.  It 
has  a  wide  range  of  frequency 
of  interruption  and  when  in 
good  working  order  can  be 
depended  upon  to  give  a  con-  Fig.  1. 

stant  root-mean-square,  cur- 
rent in  the  secondary  of  the  induction  coil.  C  is  a  condenser  the  plates 
of  which  are  separated  by  sheets  of  glass  which  can  resist  differences  of 
potential  of  30,000  volts  or  more.  This  condenser,  when  charged  by 
the  induction  coil  to  the  point  of  breaking  down  the  spark-gap  S,  dis- 
charges through  the  variable  inductance  L. 

Possibly  the  greatest  difficulty  in  maintaining  the  discharge  of  a  con- 
denser with  sufficient  constancy  of  current,  to  enable  one  to  use  it  for 
precise  measurements,  lies  in  the  spark-gap  used.  A  long  series  of  ex- 
periments showed  conclusively  that  the  electrodes  must  be  clean  and 
bright,  and  separated  by  an  unvarying  distance.  Zinc  electrodes,  which 
are  suitable  for  some  purposes,  are  soon  covered  by  an  oxide  which  causes 
the  width  of  the  gap  to  vary.  Fine  jets  of  mercury  were  finally  selected 
as  the  electrodes.  The  jets  were  perpendicular  to  each  other  in  a  hori- 
zontal plane  and  at  the  point  of  crossing  were  about  a  millimeter  apart 
and  half  a  millimeter  in  diameter.  Their  surfaces  remained  always  clean 
and  bright  and  the  distances  between  them  seemed  to  be  unvarying.  With 
these  arrangements  oscillating  currents  were  obtained,  adjustable  in 
frequency  from  twenty  thousand  to  ten  millions  or  more  cycles  per 
second,  and  sufficiently  uniform  for  precise  measurements. 

(b)  Measurement  of  Intensity  of  Current. — In  measuring  the  root- 
mean-square  intensity  of  high  frequency  currents,  it  was  found  that 
several  precautions  were  necessary.  An  instrument  was  first  made  which 
consisted  of  two  parallel  bars,  each  seven  centimeters  long,  one  centimeter 


HIRAM  WHEELER  EDWARDS. 


[VOL.  XXXIII. 


wide  and  three  millimeters  thick.  The  bars  were  connected  by  seven 
No.  40  B.  &  S.  gauge  high  resistance  wires,  placed  one  centimeter  apart. 
A  nickel-iron  thermo-couple  was  soldered  to  the  middle  point  of  each 
wire  and  was  properly  calibrated  by  direct  current.  The  alternating 
current  leads  were  connected  to  the  ends  of  the  bars  from  the  same  side. 
It  was  found  that  with  a  direct  current  flowing  through  the  instrument 
each  resistance  wire  conducted  one  seventh  of  the  total  current.  An 
alternating  current  of  frequency  about  300,000  was  sent  through  the 
instrument  and  readings  were  taken  from  the  thermo-couples.  It  was 
found  that  the  ratio  of  currents  in  the  two  extreme  wires  was  about 
seven  to  nine,  with  the  wire  nearest  the  leads  carrying  the  most 
current.  This  unequal  distribution  shows  the  necessity  for  caution  in 
the  constuction  of  an  ammeter  that  is  to  be  used  for  alternating  currents 
of  high  frequency. 

To  eliminate  the  danger  of  unequal  current  distribution  an  ammeter 
was  made  as  is  shown  in  Fig.  2.  In  place  of  the  two  parallel  bars  in  the 

instrument  described  above, 
there  were  substituted  two 
triangular  blocks  of  copper, 
PI  and  PI,  with  their  par- 
allel edges  about  three  cen- 
timeters apart.  The  bases 
of  the  triangles  measured  4 
cm.  and  the  altitudes  10  cm. 

The  blocks  were  connected  by  fifteen  no.  40  resistance  wires,  spaced 
about  2  mm.  apart.  On  the  center  wire  was  soldered  a  thermo-couple 
T.C.,  which  is  connected  to  a  sensitive  D'Arsonval  galvanometer  G. 
Resistance  wires  as  small  as  no.  40  were  used  so  that  one  could  be  sure 
that  the  alternating  current  resistance,  for  frequencies  up  to  1,000,000, 
was  not  noticeably  different  from  the  direct-current  resistance,  and  hence 
the  instrument  could  be  calibrated  by  direct  current.  For  currents 
ranging  between  one  and  two  amperes,  fifteen  resistance  wires  gives 
about  the  best  sensitiveness.  This  form  of  ammeter  may  be  used  for  a 
very  wide  range  of  current  strength  by  selecting  a  proper  number  of 
resistance  wires. 

In  using  the  latter  instrument  to  obtain  data  for  this  paper  each 
reading  of  alternating  current  was  calibrated  by  direct  current  within  a 
few  minutes  after  the  alternating  currents  had  passed  through  it.  This 
eliminates  the  danger  of  incorrect  readings  caused  by  a  change  in  room 
temperature.  A  wooden  box  protected  the  instrument  from  any  air- 
drafts. 


Fig.  2. 


No.  3.]  THE  DISTRIBUTION  OF  CURRENT.  ]  88 

(c)  Measurement  of  Frequency  of  Alternation. — Fleming  has  devised 
an  instrument  for  measuring  the  frequency  of  alternating  currents  which 
he  calls  a  photographic  spark  counter.1  Since  it  has  been  modified  in 
some  particulars  it  will  be  described  here.  There  is  first  an  enclosing 
box  about  25  cm.  square  and  40  cm.  high.  See  Fig.  3,  which  is  a  diagram 
showing  the  plan  of  the  essential  parts.  A  vertical  shaft  near  the  middle 
of  one  side  has  mounted  on  it  a  four-sided  mirror,  also  a  fly  wheel  to 
insure  uniformity  of  rotation.  A  collimator  tube  projects  from  one 
side  of  the  box  in  line  with  the  mirror.  The  spark-gap  is  placed  directly 
in  front  of  the  collimator  tube.  The  light  from  the  spark  passes  through 
a  lens  to  the  mirror  and  is  there  reflected  at  an  angle  of  about  ninety 
degrees  to  a  slit  n  the  side  of  the  box.  A 
suitable  plateholder  is  held  in  front  of  the  -F^  ^F--- 
slit  and  is  so  constructed  that  the  photo- 
graphic plate  may  be  raised  or  lowered  by  a 
string  passing  through  the  top  of  the  holder. 
Ordinarily  the  plate  was  about  twenty  centi- 
meters from  the  mirror.  For  the  higher  fre-  Fig.  3. 
quencies  it  was  found  necessary  to  extend 

the  box  so  that  the  plate  was  about  fifty  centimeters  from  the  mirror. 
The  mirror  was  rotated  by  a  motor  driven  by  storage  cells.  The  speed 
of  rotation  of  the  mirror  could  be  made  as  much  as  one  hundred  revo- 
lutions per  second,  but  was  ordinarily  rotated  at  a  speed  from  sixty  to 
eighty  revolutions  per  second.  A  speed  counter  attached  directly  to 
the  upper  end  of  the  mirror  shaft,  with  the  aid  of  a  stop  watch,  gave  a 
measure  of  the  angular  velocity  of  the  mirror. 

The  plate  was  raised  or  lowered  by  hand  since  it  was  not  necessary 
to  know  the  velocity  with  which  it  moved.  The  projection  of  the  train 
of  oscillations  on  the  plate  could  easily  be  made  less  than  the  width  of  the 
plate  by  properly  modifying  the  speed  of  the  motor.  The  sigma  brand 
of  Lumiere  plates  was  used  with  excellent  results.  The  average  distance 
between  images  of  sparks  was  measured  by  means  of  calipers  and  standard 
scale.  From  the  average  distance  between  images,  together  with  the 
dimensions  of  the  apparatus  involved,  the  frequency  of  the  sparks  was 
calculated.  The  accuracy  of  these  measurements  is  probably  within 
two  or  three  per  cent. 

In  early  trials  with  the  apparatus,  it  was  found  that  the  upper  limit 
of  frequency  of  sparks  easily  counted  was  about  one  million  per  second. 
At  frequencies  higher  than  this,  the  successive  images  of  individual  sparks 
were  fused  together  into  bands,  thirty  to  fifty  or  sixty  in  number,  each 
band  representing  a  complete  oscillatory  discharge  of  the  condenser,  and 

1  Loc.  cit. 


189  HIRAM  WHEELER  EDWARDS.  [VOL.  XXXIII. 

the  number  of  bands  representing  the  number  of  discharges  occurring 
in  the  well-known  multiple  manner.  With  lower  frequencies  the  bands 
were  drawn  out  into  trains  of  oscillatory  sparks.  There  were  usually 
from  twelve  to  eighteen  complete  oscillations  in  each  train  visible  on  the 
plate.  Since  the  intensity  of  luminosity  of  the  spark  becomes  gradually 
weaker,  it  is  impossible  to  say  how  many  oscillations  there  are  in  each 
train,  for  the  sensitiveness  of  the  plate  limits  the  number  of  those  that 
can  be  counted.  The  character  of  the  photographs  has  been  shown  by  a 
number  of  previous  writers,  among  whom  Trowbridge1  has  done  extensive 
work. 

(d)  The  Differential  Electric  Thermometer. — To  compare  the  virtual 
resistance  with  the  direct  current  resistance,  a  differential  electric  ther- 
mometer was  used.  This  instrument  was  devised  and  used  by  Fleming.2 
As  shown  in  Fig.  I  it  consists  of  two  glass  tubes,  T\  and  T<L,  each  about 
125  cm.  long  and  4  cm.  in  diameter.  They  are  as  nearly  alike  as  possible. 
A  small  tube  with  a  bore  about  a  millimeter  in  diameter  connects  the 
two  larger  tubes  which  are  otherwise  air-tight.  The  wires  under  test 
pass  through  rubber  stoppers  at  the  ends  and  complete  the  electric 
circuit  through  either  S\  or  Sz,  which  are  highly  insulated,  six-pole 
switches.  These  are  used  to  connect  the  wire  in  either  tube  with  the 
direct  current  or  the  oscillating  current  as  may  be  desired.  A  small  drop 
of  ether  or  some  other  light  liquid  in  the  capillary  tube  will  indicate  any 
differences  of  heat  generation  in  the  tubes  T.  It  is  possible  to  balance 
the  heat  developed  by  an  alternating  current  in  one  tube,  by  an  equal 
amount  of  heat  in  the  other  tube.  Since  the  heat  produced  in  any  circuit 
is  proportional  to  the  resistance  and  the  square  of  the  current,  at  the 
point  of  balance  the  ratio  of  resistances  may  be  expressed  as  the  inverse 
ratio  of  the  squares  of  the  currents.  From  the  ammeters  in  the  direct- 
current  circuit  and  the  alternating-current  circuit  the  values  of  the  two 
currents  are  read,  and  from  these  two  readings  the  ratio  of  resistances  is 
calculated.  In  order  to  minimize  any  error  which  might  enter  into  the 
measurements,  because  of  differences  of  specific  heats  of  the  glass  in 
the  two  tubes,  or  in  the  amount  of  glass,  or  radiating  power,  the  currents 
in  the  wires  were  interchanged  a  number  of  times  and  the  average  results 
taken. 

It  was  necessary  to  arrange  both  the  alternating-current  circuits,  one 
passing  through  7\  and  the  other  through  T2,  so  that  in  interchanging 
circuits  during  the  course  of  a  particular  set  of  readings,  there  would  be 
no  difference  of  frequency.  This  was  done  by  changing  the  area  enclosed 
by  one  circuit  until  the  frequency  of  the  current  in  that  circuit,  as 

1  Trowbridge,  Phil.  Mag.,  August,  1894,  p.  182. 

2  Loc.  cit. 


No.  3-]  THE  DISTRIBUTION  OF  CURRENT.  1 90 

measured  by  means  of  the  photographic  plate,  was  the  same  as  the 
frequency  of  the  current  when  passing  through  the  other  circuit. 

In  taking  any  set  of  readings  for  a  definite  frequency,  the  method  of 
procedure  was  to  balance  three  or  four  different  values  of  alternating 
current  by  direct  current,  with  each  value  of  alternating  current  passing 
through  both  circuits  T\  and  TV  The  average  of  these  six  or  eight 
readings  was  used  to  calculate  the  ratio  of  Rf  to  R. 

The  oscillating  spark  was  photographed  by  the  spark  counter  during 
one  of  these  readings.  The  average  distance  between  images  on  the 
photographic  plate  was  taken  from  three  or  four  trains  of  images  and 
this  average  distance  used  in  calculating  the  frequency.  The  angular 
velocity  of  the  mirror  could  be  measured  within  two  or  three  per  cent. 
The  frequency  of  the  oscillations  could  probably  be  calculated  within 
three  or  four  per  cent,  of  the  correct  value.  The  error  in  the  meas- 
ured ratio  of  Rr  to  R  is  not  more  than  four  per  cent,  of  the  correct  value. 

(e)  Test  of  the  Apparatus. — In  order  to  test  the  efficiency  of  the  appa- 
ratus, cylindrical  wires  were  introduced  into  the  differential  thermometer 
and  observations  made  on  the  change  of  resistance.  Another  and  more 
important  reason  for  taking  this  preliminary  set  of  readings,  was  to 
discover  whether  the  damping  of  the  oscillations  was  great  enough  to 
cause  any  appreciable  difference  between  the  observed  ratio  of  resistance 
and  that  calculated  by  a  formula  based  upon  the  assumption  that  the 
oscillations  were  undamped.  The  results  are  shown  in  Table  I.  Since 
the  experimental  readings  are  within  less  than  one  per  cent,  of  the  values 
calculated  by  Maxwell's  formula,  it  may  be  assumed  that  the  damping 
is  small  enough  to  neglect,  and  that  the  apparatus  may  be  depended  upon 
to  give  reliable  readings,  to  the  degree  of  precision  indicated. 

TABLE  I. 

Variation  of  Resistance  of  Cylindrical  Wires.     Copper  Wire,  No.  25  Brown  and  Sharpe  Gauge, 

0.045  cm.  in  Diameter. 

Frequency.                            R'lR  Observed.  R')R  Calculated. 

36,100                                        1.000  1.001 

138,000                                      1.062  1.061 

283,000                                      1.215  1.213 

313,000                                      1.242  1.244 

Maxwell's  formula  used  in  calculating  R'jR  in  the  above  table  is  as 
follows : 

Rf  _ 

R  ~         ' 

where  N  is  the  frequency,  d  is  the  diameter  of  the  wire  and  p  is  the 
specific  resistance.  For  the  wire  used  above  d  =  0.045  cm-  and  p  was 
taken  at  1,600. 


HIRAM  WHEELER  EDWARDS. 


[VOL.  XXXIII. 


To  illustrate  how  the  observed  values  of  R'/R  were  obtained  the  fol- 
lowing data  are  quoted  for  the  case  where  the  frequency  was  283,000. 
At  the  point  of  balance  in  the  differential  electric  thermometer: 

Avg.  direct  current  through  TI  =  1.725  amps. 


Avg.  alt. 
Avg.  direct 
Avg.  alt. 


T2  =  1-535 
T2  =  1.670 
T!  =  1.546 


KIK  -  (£§:)'  -  ••"• 

Avg.  R'/R  =  1.215. 
The  frequency  was  obtained  from  the  following  formula : 


N  —  47rcor//  =  283,000. 

where  w,  the  speed  of  the  mirror,  =  70.4  revolutions  per  second, 

r,  the  distance  from  the  mirror  to  the  plate,  =  53.5  cm.  and 

/,  the  average  distance  between  images  on  the  plate,  =  0.167  cm- 

(/)  The  Virtual  Resistance  of  Wires  of  Square  and  Rectangular  Cross- 
Section. — In  applying  the  formulas  no.  31  and  no.  27  (developed  below) 
to  the  experimental  results  given  in  Fig.  4,  it  will  be  assumed  that  the 
effects  of  damping  of  the  oscillating  current  are  negligible.  Formula 
(27)  is  more  useful  for  calculating  the  ratio  of  apparent  resistance  to  that 
offered  to  direct  current,  in  the  case  of  alternating  currents  of  lower 
frequency. 

The  calculated  values  of  the  ratio  of  the  virtual  resistance  to  the 


•*•" 

^ 

f*G 

^ 

**• 

^ 

0. 

=i 

= 

— 

.-* 

•— 

K 

ja 
10 

— 
0( 

.—  * 

JO 

I 

'ij 

r_ 

?0 

4 

0 

,. 

0( 

JO 

3C 

"o 

0^ 

)0 

N. 

direct-current  resistance  for  the  wire  of  which  2a  =  0.059  cm->  were 
plotted  as  shown  in  Fig.  4.     The  curve  is  drawn  from  the  calculated 
values  and  the  small  circles  indicate  the  experimental  observations. 
Fig.  5  shows  in  a  similar  manner  the  nearness  with  which  the  experi- 


100.000.  200,000. 

Fig.  5. 


300,000.   A/. 


No.  3.] 


THE  DISTRIBUTION  OF  CURRENT. 


I92 


mental  values  check  the  calculated  ones  for  the  square-sectioned  wire  of 
which  2a  =  0.070  cm. 

Replacing  the  square  wires  by  wires  of  rectangular  section,  another 
series  of  observations  was  obtained,  the  results  of  which  are  shown  in 
Fig.  6  and  Fig.  7.  The  calculated  values  of  R'/R  for  the  wire  of  which 


i  00,000.  200,000. 

Fig.  6. 


300,000.   H. 


2a  =  0.039  cm-  and  2b  =  0.0665  cm->  were  used  in  drawing  the  curve 
of  Fig.  6,  the  small  circles  indicating,  as  above,  the  experimental  readings. 


100,000.  2001000. 

Fig.  7. 


300,000.  N 


Similarly  Fig.  7  was  drawn  for  the  wire  of  which  20,  =  0.0485  cm.  and 
2b  =  0.127  cm-  With  a  wire  as  large  as  this  last  one,  the  formula  does 
not  give  values  which  agree  satisfactorily  with  the  experimental  values 
if  the  frequency  is  much  greater  than  150,000.  Formulas  (26)  and  (30) 
were  used  here  in  calculating  the  values  for  R'/R  (see  below). 


3.  MATHEMATICAL  DEVELOPMENT. 

The  conductor,  rectangular  in  cross-section,  is  so  long  that  end  effects 
may  be  neglected.  Other  conductors  are  so  far  removed  that  their 
influence  need  not  be  considered.  Referring  to  Fig.  8,  a  reference  system 
is  chosen,  with  the  origin  at  the  center  of  any  cross-section,  the  X  and  Y 
axes  parallel  to  the  sides  of  the  section,  and  the  Z  axis  coinciding  with 
the  axis  of  the  wire.  Since  the  conductor  is  long,  it  may  be  assumed 
that  no  component  of  current  flows  parallel  to  the  XY  plane.  The  fol- 
lowing equation  then  holds: 

=  ~to*  +  W  '  (l) 


193 


HIRAM  WHEELER  EDWARDS, 


[VOL.  XXXIII. 


—^b 


2 d 


where,  in  agreement  with  Max- 
well's notation,  w  is  the  z  compo- 
nent of  current  intensity,  H  is  the 
z  component  of  vector  potential  of 
magnetic  induction  and  ^  is  the 
permeability  of  the  material  of  the 
conductor. 

The  impressed  electromotive  in- 
tensity may  be  supposed  equal  at 
all  points  of  the  cross-section  of  the 
wire,  assuming  that  the  specific 

resistance  is  uniform.  Let  the  electromotive  force  per  unit  length  be  Ee^nt. 
Considering  only  instantaneous  relations,  the  total  electromotive  in- 
tensity, inductive  and  non-inductive,  in  the  wire  is 


P  =  E  - 


dH 

dt 


(2) 


Outside  the  wire  the  corresponding  electromotive  intensity  is 

dH 
P=-.T-  (3) 


If  the  specific  resistance  of  the  wire  is  p  and  the  specific  inductive  capacity 
of  the  medium  is  K 

(4) 


Considering  the  inductive  action  only, 


dt' 


and  eliminating  w  between  equations  (i)  and  (4) 


(5) 


(6) 


In  the  wire  the  current  may  be  regarded  as  due  to  the  conductivity  alone, 
and  if 

then 

19  TT  n9TT 

-  o.  (7) 


In  the  surrounding  medium  i/p  may  be  neglected  and  putting  h  =  n/V, 


No.  3-1  THE  DISTRIBUTION  OF  CURRENT.  1 94 


where  F,  the  velocity  of  propagation  of  the  disturbance,  is  \\V Kp, 


The  general  solution  of  (7)  is 

H  =  A^+B^+CF+Dr*  +  Fe"^+Mr*^+Ne"^+Qr*^*,  (9) 
where  k'  =  fc/1/2,  and  if  h'  =  h/l/2  the  solution  of  (8)  is 

_j_  jyjh'x^y  _|_  Q'e-ih'^=-y  ^        ( l  °) 

By  symmetry  relations  the  value  of  H  must  remain  unchanged  if  +  x  is 
written  for  —  x  and  +  y  for  —  y  in  the  equations  (9)  and  (10).  This 
shows  at  once  that  A  =  B,  A'  =  B',  C  =  D,  C'  =  D't  F  =  M  =  N  =  Q 
and  F'  =  M'  =  N'  =  Q' .  Hence  inside  the  wire 

H=2A  cosh  kx+2C  cosh  ky+2F  (cosh  k'x+y+  cosh  k'x—y),       (n) 
and  outside 


#=2^4'  cosh  ^wc+2C'cosh  ihy+2F'  (coshih'x+y  +  cosh  ih'x  —  y).     (12) 

If  the  wire  has  a  square  cross-section  instead  of  a  rectangular  section 
then  x  can  be  interchanged  with  y  in  (n)  and  (12)  without  affecting  H. 
This  necessitates  the  additional  simplification  that  A  =  C  and  A'  =  C'. 

To  evaluate  the  constants  of  (n)  and  (12),  the  continuity  of  the  total 
electromotive  intensity  and  magnetic  field  intensity,  at  the  boundary,  is 
used.  If  a  and  |8  are  the  x  and  y  components  of  magnetic  field  intensity, 
then  since  the  curl  of  the  vector  potential  is  the  magnetic  field  intensity, 
at  the  point  x  =  b,  y  =  o,  by  the  continuity  of  /3  the  following  relation 
is  obtained  from  (n)  and  (12): 

kA  sinh  kb  +  2k' F  sinh  k'b  =  ihA'  sinh  ihb  +  2ih'F'  sinh  ih'b.  (13) 
Similarly  by  the  continuity  of  a  at  x  =  o,  y  =  a, 

kC  sinh  ka  +  2k' F  sinh  k'a  =  ihC'  sinh  iha  +  2ih'Ff  sinh  ih'a.  (14) 
Also  at  the  point  x  =  b,  y  =  a,  for  both  a  and  ]S, 


sinh  kb  +  k'F  (sinh  k'b  +  a  +  sinh  jfe'&  -  a)  =  UL€'  sinh  $* 

+  ih'F'(sinh ih'b  +  a  +  sinh  tfc'6  -  a), 


kC  sinh  j^a  +  k'F  (sinh  k'b  +  a  -  sinh  fc'6  -  a)  =  tAC'  sinh 


+  ih'F'(&nhih'l+a  -  sinh  ih'b  -a). 


195  HIRAM  WHEELER  EDWARDS.  [VOL.  XXXIII. 

Also  by  the  continuity  of  the  total  electromotive  intensity  at  x  =  b 
y  =  o  and  at  x  =  o,  y  =  a  using  equations  (2)  and  (3),  two  more  relations 
may  be  obtained: 


E  —  2inAcoshkb  —  4inFcoshk'b  =  —  2inA'coshihb  —  2inFfcoshihfb,  (17) 
E  —2inCcoshka—4.inFcoshk'a  =  —  2inC'coshiha  —  2inF'coshih'a.  (18) 


From  equations  (13)  to  (18)  the  values  of  the  constants  may  be  found. 
The  total  current  passing  through  any  cross-section  may  be  found  by 
taking  the  line  integral  of  the  magnetic  field  intensity  around  the  bound- 
ary.    If  7  is  the  total  current 

7  =  4  I    Pdy  -  4  I    adx.  (19) 

Jo  Jo 

Inserting  the  values  of  a  and  /3  in  this  equation  and  integrating  gives 
the  desired  electromotive  force  equation,  from  which,  expressions  for 
the  virtual  resistance  and  inductance  may  be  calculated.  The  determina- 
tions of  the  constants  by  equations  (13)  to  (18)  is  so  complicated  as  to 
be  unmanageable.  It  is  possible,  however,  to  obtain  approximate  formu- 
las with  mean  value  for  the  constants  by  assuming  that  A  =  C  =  F  in 
equation  (n)  and  neglecting  the  effect  of  the  medium.  The  assumption 
that  A  =  C  means  that  the  vector  potential  along  the  x  axis  is  the  same 
as  along  the  y  axis  for  equal  values  of  the  argument.  This  is  true  for  a 
square  sectioned  wire  but  when  considering  a  rectangular  sectioned  con- 
ductor it  is  only  approximately  true,  the  degree  of  approximation  depend- 
ing upon  the  difference  between  the  two  dimensions.  The  justification 
for  putting  A  —  F  is  based  upon  the  substantiation  of  the  results  as 
calculated  for  particular  cases  from  experimental  observations,  The 
values  of  R'/R  for  four  particular  cases,  calculated  by  a  formula  based 
upon  this  assumption  are  given  in  Figs.  5—8.  Since  the  observed  values 
agree  up  to  frequencies  of  150,000,  writh  these  calculated  values,  it  must 
be  that  up  to  this  limit  F  is  nearly  equal  to  A  .  For  the  two  rectangular 
sectioned  wires  this  limit  is  reached  at  frequencies  of  150,000  and  then 
the  formulas  fail,  then  the  differences  between  the  observed  and  calcu- 
lated values  increases  as  the  frequency  increases.  For  the  larger  square 
sectioned  wire  the  values  agree  within  five  per  cent,  up  to  a  frequency 
of  150,000,  and  then  the  differences  increase  to  about  ten  per  cent,  at  a 
frequency  of  265,000,  and  at  312,000  the  values  are  again  in  close  agree- 
ment. The  differences  for  the  smaller  square  sectioned  wire  are  not 
greater  than  three  per  cent,  for  the  range  tested,  namely  up  to  a  fre- 
quency of  257,000. 


No.  3-1  THE  DISTRIBUTION  OF  CURRENT.  196 

To  obtain  these  approximate  expressions  equation  (7)  will  be  used,  but 
put  into  more  convenient  form.  Introducing  the  non-inductive  part 
of  the  electromotive  intensity,  which  may  be  called  d\I//dz,  into  equation 
(4)  and  neglecting  the  effect  of  the  medium,  gives 

dt       dH 
pW=-dz--V'  (20) 

Eliminating  w  between  this  equation  and  equation  (i) 
d*H      d*H          i          idf\ 

VT  +  irir  -  k2 1 H  -  -   -  ]  =  o.  (21) 

dx2       dy2  \  n  dzJ 

i  d\L> 
Since-  —  is  independent  of  x  or  y,  (21)  may  be  written 

Yl    uZ 


(22) 


The  general  solution  of  this  equation  is  similar  to  the  solution  of  (7) 
and  has  eight  undetermined  coefficients.  This  number  may  be  reduced 
to  three  by  the  symmetry  relations  as  is  shown  above.  Assuming  now 
that  these  three  constants  are  all  equal  leads  to  the  following  solution: 


H  =  -  ,    +2A  (cosh  kx+cosh  ky+cosh  k'x+y+caah  k'x-y).    (23) 

fl  oZ 

The  value  of  the  single  undetermined  coefficient  may  be  found  by 
integrating  the  expression  for  current  intensity  over  the  surface  of  the 
cross-section.  If  7  is  the  total  current, 

7  =  4    I      I    wdydx. 

Jo    Jo 

Using  the  value  of  w  as  determined  by  equations  (i)  and  (23),  with  the 
aid  of  the  above  integral,  the  value  of  A  may  be  shown  to  be 


2(ka  sinh  kb-}-kb  sinh  ka-\-2  cosh  k'b+a  —  2  cosh  k'b—a) 

If  at  the  point  x  =  b,  y  =  o,  H  is  put  equal  to  some  constant,  say  L, 
times  the  current  and  if  /  be  the  length  of  the  conductor  considered, 

d\l/ 
multiplying  equation  (23)  by  tin  and  putting  —  /  —  =  E,  the  electro- 

motive force,  gives 
E  =  Lliny 

cosh  kb+2  cosh  k'b+i 


ka  sinh  kb+kb  sinh  ka+2  cosh  k'b+a—2  cosh  k'b—a 


(25/ 


197  HIRAM  WHEELER  EDWARDS.  [VOL.  XXXIII. 

If  the  hyperbolic  functions  are  expanded  in  series  of  powers  of  k 
and  the  numerator  divided  by  the  denominator,  the  following  expression 
is  obtained  : 


\ 

(26) 


45PZ 
_  7rVJV4(-i72fr8  +  io7ofr6a2+28o76V-34o62a6- 

14175'P4  l"*"       J 

where  R  =  Ip/^ab  (the  resistance  of  a  length  /  for  direct  current)  and 
N  =  n/2-ir,  the  number  of  alternations  per  second.  Writing  R'  for  the 
virtual  resistance,  gives 


R'  _  _ 

~R  =  45P2  (27) 

If  the  wire  has  a  square  cross-section  the  expression  for  the  virtual 
resistance  may  be  found  by  putting  b  =  a  in  (26) 


R'  ,  , 

R  -  45P^"  I4I75  -P4 


Formulas  (26)  and  (27)  may  be  used  in  calculating  R'/R  for  particu- 
lar cases  where  the  dimensions  and  frequency  are  of  a  magnitude  which 
will  give  a  value  of  0.5  or  less  to  the  second  term  of  the  series.  If  the 
frequency  and  dimensions  are  large  then  the  series  is  not  rapidly  con- 
vergent. The  calculation  of  more  terms  of  the  series  is  troublesome. 
To  obtain  a  complete  expression  the  fraction  in  (25)  may  be  expressed  in 
exponential  functions  and  then  separated  into  real  and  imaginary  parts. 
The  real  coefficient  of  7  is  then  the  desired  expression  for  R'.  The 
device  used  in  making  the  necessary  transformation  is  embodied  in  the 
following  relations: 


/ —         I27TU72 

=  V2i  \  — 

*       n 


where 


5=J2-^.  (28) 


Hence    each  hyperbolic  function  may  be  expressed  in  somewhat  the 
following  manner: 

cosh  kb  =  %(ekb  +  e~7cb)  =  cos  bS  cosh  bS  +  *  sin  bS  sinh  bS. 


No.  3.]  THE  DISTRIBUTION  OF  CURRENT.  .      198 

Making  these  substitutions  in  (25)  and  then  rationalizing  the  denomina- 
tor of  the  fraction,  gives 

Olid  +  £iA)  +  i(£iCi  -  AlDl)   . 
E  =  Lhnj  +  Hmrmr  -  —   2nrir~  ~  »  29) 


where 

^4i  =  cos  65  cosh  bS  +  2  cos        cosh  —  _-  +  i  , 

1/2  1/2 

£1  =  sin  bS  sinh  65+2  sin  —  —  sinh  —  —  , 

1/2  1/2 

Ci  =  05  cos  65  sinh  65  —  aS  sin  65  cosh  65  +  65  cos  a5  sinh  aS 
—  65  sin  aS  cosh  a5+2  cos  —  =  S  cosh  —  -5—2  cos  —  =  5  sinh  —  =•  5, 

1/2  1/2  1/2  F2 

Di  =  aS  cos  65  sinh  65  +  a5  sin  65  cosh  65  +  65  cos  aS  sinh  a5 

+  65  sin  a5  cosh  a5+  2  sin  —  =5  sinh—  -5—  2  sin—  -  5  sinh  —  :—S. 

1/2  1/2  1/2  1/2 

Hence  : 


_  ,     . 

~R=    ~T~        ~W+Df~ 

For  the  wire  of  square  cross-section  this  expression  becomes  simplified 
and 

R'  _  4>jrn»a2      A2D<2  -  £2C2  ,     , 

j?  ~  *      r-2  j_  n  2    '  ^  ' 

J<  P  ^  -r  -^2^ 

where  the  change  in  subscripts  indicate  the  changes  in  the  constants 
which  are  obtained  by  putting  6  =  a. 

It  is  shown  above  that  these  expressions  can  be  used  to  calculate  the 
virtual  resistance  within  certain  limits. 

Before  leaving  the  mathematical  side  of  the  investigation  an  expres- 
sion for  finding  the  intensity  of  current  at  any  point  of  the  cross-section 
of  the  conductor  is  to  be  developed.  For  present  purposes  a  study  of 
the  distribution  of  current  in  a  wire  of  square  cross-section  is  just  as 
instructive  as  that  in  a  wire  of  rectangular  cross-section. 

Using  equation  (i)  for  current  intensity  and  (23)  for  vector  potential 
the  following  expression,  for  the  wire  of  square  section,  is  obtained: 


J  99  HIRAM  WHEELER  EDWARDS.  [VOL.  XXXIII. 

The  exponential  functions  of  this  expression  may  be  separated  into  real 
and  imaginay  parts  by  a  method  of  procedure  similar  to  the  method  used 
above. 


us-  -r  -3  '  (33 

where 

AS  =  —  I  sin  Sx  sinh  Sx-{- sin  ^S-y  sinh  .Sy+sin  — —  S  sinh  — —  5 

1/2  V  2 

+  sin  — —S  sinh  - 

V2  1/2 

x  ~\~  y  x  ~\~  y 

BS  =  cos  Sx  cosh  Sx  +  cos  .Sv  cosh  •S'y  +  cos  — -=-  S  cosh  — ^—  S 

1/2  1/2 

+  cos  — —  S  cosh  — —  5, 

V2  V2 

Cs  =  cos  1/2  a5  cosh  1/2  a.S+^-S'  cosh  aS  sinh  aS  —  aS  sin  a,S  cosh  a5  —  2, 
Dz  =  sin  1/2  a^1  sinh  1/2  #5  +  aS  cos  aS  sinh  a5  +  aS  sin  a5  cosh  a5. 


The  equation  (33)  expressing  current  intensity  at  any  point  in  the 
cross-section  of  the  wire  is  like  the  general  electromotive  force  equation 
in  that  it  is  a  complex  quantity.  The  form  must  be  complex  because  part 
of  the  current  is  caused  by  the  inductive  action  and  the  other  part  by 
the  non-inductive  action.  If  equation  (33)  were  multiplied  by  p,  the 
specific  resistance,  the  resulting  equation  would  be  an  electromotive 
force  equation,  which  is  in  general  complex. 

To  obtain  values  of  w  at  points  in  the  cross-section  of  a  particular 
wire  under  given  conditions,  the  real  and  imaginary  parts  must  be 
calculated  separately,  and  then  added  by  vector  methods,  or  analyti- 
cally by  finding  the  square  root  of  the  sum  of  their  squares. 

4.  VARIATION  OF  CURRENT  INTENSITY. 

Another  purpose  of  this  paper  is  to  show  the  variation  of  current 
intensity  or  density  of  flow,  over  the  sectional  area  of  a  long,  straight 
conductor  of  square  cross-section,  carrying  current  of  high  frequency. 
The  mathematical  development  leads  to  a  computation  of  current  intensity 
at  any  point  of  the  section.  Since  no  experimental  method  of  measuring 
this  quantity  has  thus  far  been  devised,  the  equations  derived  from  theo- 
retical considerations  will  be  relied  upon  entirely.  The  justification  for 
this  course  lies  in  the  satisfactory  agreement  between  experimental  and 
theoretical  results,  already  described  in  another  part  of  this  paper.  It 
appears  that  the  experimental  difficulties  in  the  measurement  of  current 


No.  3.]  THE  DISTRIBUTION  OF  CURRENT.  2OO 

intensity  would  be  considerable.  If  the  wire  be  separated  into  filaments, 
like  the  fine,  parallel  wires  of  a  flexible  cable,  the  conditions  which  cause 
the  uneven  distribution  of  current  have  been  modified  to  such  an  extent 
that  the  measurement  of  current  in  each  filament  would  be  of  little  value 
in  solving  the  problems  proposed.  Equation  (33)  will  be  used  in  the 
calculations  which  follow. 

A  wire  or  bar,  two  centimeters  square,  has  been  selected  for  illustration. 
To  simplify  the  calculation  let  the  frequency  be  4OO/7T2.  This  makes  5"  of 
equation  (33)  unity.  Since  the  wire  is  large  in  cross-section,  even  with 
low  frequency  the  variation  in  current  intensity  will  be  similar  to  the 
variation  in  a  smaller  wire  with  higher  frequency.  The  similarity  may 
be  seen  from  an  inspection  of  formula  (32).  It  will  be  noticed  that 
wherever  V  N  appears  in  the  equation  as  a  factor,  a  dimensional  length 
Xj  yora  appears  also  as  a  factor.  N  appears  elsewhere  only  in  K2.  If 
V  Nd  (where  d  is  a  dimension)  is  put  equal  to  some  constant,  then  I/ AT" 
and  d  may  be  varied  individually,  so  long  as  their  product  remains 
constant,  and  the  resulting  value  for  w  will  be  proportional  to  the 
frequency. 

Consider  the  square  section  as  a  special  case  of  the  rectangular  section 
of  Fig.  8  in  which  b  equals  a.  Fig.  9  shows  graphically  the  variation  of 
current  intensity  along  the  three  principal  lines  of  the  particular  square 
selected.  Curve  A  represents  the  variation  from  the  origin  along  the 
line  y  =  o  to  x  =  =*=  a.  Curve  B  shows  the  variation  from  the  middle 
of  any  side  on  either  axis  along  the  edge  to  the  vertex.  This  curve 
represents  the  symmetrical  variation  of  current  along  eight  different 
lines  of  the  square.  In  curve  C  the  variation  of  current  intensity 
along  a  diagonal  from  the  origin  to  any  one  of  the  four  vertices  is 
shown. 

Another  graphical  representation  of  the  density  of  current  flow  over 
the  section  of  the  same  square  wire,  at  the  same  frequency  of  alternation, 
is  made  in  Fig.  10.  Here  contour  lines  are  drawn  for  one  quadrant  of 
the  section,  the  difference  between  consecutive  lines  being  0.005  C.G.S. 
unit  of  current.  The  line  nearest  the  origin  indicates  an  intensity  of 
0.215  C.G.S.  unit.  The  current  is  fairly  uniform  over  the  central  region, 
but  proceeding  from  the  origin  along  a  diagonal  to  a  vertex  the  lines  are 
drawn  more  closely  together.  It  might  be  supposed  at  first  sight  that 
for  the  case  of  very  high  frequency  the  current  would  be  localized  entirely 
at  the  corners.  That  such  is  not  the  case  may  be  seen  by  inspection  of 
formula  (33).  Putting  x  and  y  both  equal  to  zero,  the  numerator  still 
contains  N,  which,  although  small  in  comparison  with  the  denominator, 
gives  w  a  value  different  from  zero. 


2OI 


HIRAM  WHEELER  EDWARDS. 


[VOL.  XXXIII. 


In  any  electromagnetic  phenomenon  the  law  of  the  conservation  of 
energy  is  valid.     In  circuits  such  as  described  above  the  energy  is  mani- 


Fig.  9. 


Fig.  10. 


fested  in  two  ways,  namely,  in  the  production  of  heat  and  in  the  produc- 
tion of  the  magnetic  field.  There  would  be  a  minimum  of  heat  produced 
if  the  current  were  distributed  uniformly  over  the  cross-section.  The 
energy  of  the  magnetic  field  would  be  a  minimum  if  the  current  were 
entirely  on  the  surface  of  the  wire.  Since  both  of  these  conditions  can- 
not be  satisfied  simultaneously,  a  compromise  is  established  which  makes 
the  total  energy  a  minimum.  This  compromise  is  shown  for  a  particu- 
lar case  in  Fig.  n. 

To  show  the  variation  in  current  intensity  for  different  frequencies, 
also  to  show  that  the  variation  in  a  small  wire  with  high  frequency  is 
similar  to  the  variation  in  a  large  wire  with  low  frequency,  the  values  of 
w  for  a  wire  having  2a  =  0.070  cm.,  with  a  frequency  of  101,320,  have 
been  calculated  along  the  line  corresponding  to  curve  A  of  Fig.  9  and 
are  shown  in  Fig.  n.  The  increase  in  intensity  from  the  smallest  to 
the  greatest  value  in  the  smaller  wire  is  greater  than  in  the  larger  wire, 
but  the  characteristics  of  the  two  curves  are  the  same.  A  value  of  the 
frequency  could  be  found  that  would  make  the  increase  the  same  in 

both  wires. 

The  effect  of  increasing  the 
frequency  of  alternation  of  cur- 
rent in  the  same  wire  of  square 
section  is  illustrated  by  the  two 
curves  of  Fig.  n.  Both  curves 
are  for  the  line  y  =  o  from  the 
origin  to  x  =  a.  The  length  of 
one  side  of  the  section  is  20,  = 
0.070  cm.  Curve  A  shows  the 
variation  for  a  frequency  of  101,320,  and  curve  B  for  a  frequency  of 


No.  3.]  THE  DISTRIBUTION  OF  CURRENT. 


1,000,000.  The  wire  is  supposed  to  carry  the  same  total  current  in 
both  cases,  ten  amperes.  The  effect  of  increasing  the  frequency  is  shown 
in  a  crowding  of  the  current  toward  the  boundaries  of  the  wire. 

SUMMARY. 

1.  Alternating  currents  have  been  produced  by  the  discharge  of  a 
condenser  through  an  inductive  resistance,  with  sufficient  uniformity 
of  current  to  render  possible  their  use  in  precise  measurements. 

2.  An  ammeter  is  described  which  is  suitable  for  the  measurement  of 
the  intensity  of  alternating  currents. 

3.  The  frequency  of  alternating  currents  has  been  measured  by  a 
photographic  spark  counter  for  any  frequency  up  to  three  quarters  of  a 
million  per  second. 

4.  By  checking  Maxwell's  formula  for  the  virtual  resistance  of  cylin- 
drical wires,  it  is  shown  that  the  damping  of  the  oscillating  currents  used 
is  small  enough  to  neglect. 

5.  The  ratio  of  the  virtual  to  the  direct  current  resistance  for  copper 
wires  of  square  and  rectangular  cross-section  was  measured  by  the  use 
of  a  differential  thermometer,  and  these  observed  values  were  checked 
by  approximate  formulas  within  certain  limits.     For  the  two  square 
cross-sectioned  wires  tested  this  limit  was  for  frequencies  above  one 
hundred  and  fifty  thousand  and  for  the  two  rectangular  cross-sectioned 
wires  the  limit  was  about  one  hundred  and  fifty  thousand. 

6.  Formulas  are  developed  by  certain  assumptions  in  the  general 
theory,  which  give  approximate  methods  of  calculating  the  ratio  of  the 
virtual  to  the  direct  current  resistance.     The  validity  of  these  assump- 
tions is  based  upon  experimental  results. 

7.  In  the  development  of  the  expression  from  which  the  ratio  of 
resistances  is  calculated,  there  occurs  an  expression  for  current  intensity 
at  any  point  of  the  cross-section  of  the  conductor.     This  expression  is 
used  to  calculate,  for  some  particular  cases,  the  distribution  of  current 
intensity  within  the  cross-section. 

UNIVERSITY  OF  CALIFORNIA, 
May  i,  1911. 


THE  PHYSICAL  REVIEW 

A  JOURNAL  OF  EXPERIMENTAL  AN<D  THEORETICAL  PHY3ICS 


Conducted 
With  the  Co-operation  of  the 


AMERICAN  PHYSICAL  SOCIETY 

BY 

EDWARD  L.  NICHOLS 
ERNEST  MERRITT,  AND  FREDERICK  BEDELL 


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